Uniqueness of nonnegative solutions for semipositone problems on exterior domains

We consider the problem {-Delta u = lambda K(vertical bar x vertical bar)f (u), x is an element of ohm u = 0 if vertical bar x vertical bar = r(0) u -> 0 as vertical bar x vertical bar -> infinity, where lambda is a positive parameter, Delta u = div(del u) is the Laplacian of u, ohm = {x is an element of R-n; n > 2, vertical bar x vertical bar > r(0)}, K is an element of C-1 ([r(0), infinity), (0, infinity)) is such that lim(r ->infinity) K(r) = 0 and f is an element of C-1 ([0, infinity), R) is a concave function which is sublinear at infinity and f(0) < 0. We establish the uniqueness of nonnegative radial solutions when lambda is large. (c) 2012 Elsevier Inc. All rights reserved.